Proof (Intuition) Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proof (Intuition).

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The informal, intuitive sense of why a mathematical statement must be true — the "aha" that precedes and motivates a formal proof.

A chain of reasoning that convinces you something MUST be true.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A proof must eliminate all doubt by showing the conclusion follows necessarily from the hypotheses — intuition suggests what to prove; proof establishes that it IS true.

Common stuck point: A proof is not an example. Examples suggest; proofs establish.

Sense of Study hint: Write down what you know (assumptions) and what you want to show (conclusion). Then ask: 'What is one logical step I can take from the assumptions toward the conclusion?'

Worked Examples

Example 1

easy
Before writing a formal proof that 'the sum of two even integers is even,' build the intuition. Explain why it must be true, then formalise.

Solution

  1. 1
    Intuition: Even numbers are multiples of 2. Adding two multiples of 2 gives another multiple of 2 — the '2-ness' is preserved.
  2. 2
    Analogy: pairs of objects combined with pairs of objects always give pairs.
  3. 3
    Formalise: Let a=2m and b=2n. Then a+b=2m+2n=2(m+n), which is even.

Answer

a+b = 2(m+n) \text{ is even}
Proof intuition means building a convincing internal picture before writing the formal argument. The intuition guides which definitions and algebraic steps to use, making the formal proof feel natural rather than mechanical.

Example 2

medium
Build intuition for why \sqrt{2} is irrational before writing the formal proof. What is the core contradiction?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Build intuition for the statement: 'For any integer n, n(n+1) is even.' Explain informally why this must be true.

Example 2

medium
Build intuition for induction: why does proving 'P(k) \Rightarrow P(k+1)' together with P(1) establish P(n) for all n?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

logical statementconditional